Linear guide apparatus

ABSTRACT

A linear guide apparatus has a guide rail having a rolling-element rolling groove formed on a side thereof along in an axial direction; and a slider having a rolling-element rolling groove disposed so as to be opposed to the rolling-element rolling groove of the guide rail, the slider moving relatively in the axial direction of the guide rail with rolling elements rollably interposed between the two rolling-element rolling grooves, and a crowning portion disposed on the both axial ends of the rolling-element rolling groove of the slider, wherein a maximum value of sum of load in an axial direction applied on the crowing portion on the both axial ends of the slider by the rolling elements is set to be smaller than a maximum value of load in the axial direction applied on one of the crowning portions of the slider by one rolling element.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a linear guide apparatus for use inmachine tools, industrial machines, etc.

2. Description of the Related Art

A linear guide apparatus, as a linear guide for use in a machiningcenter as a machine tool, a robot as an industrial machine, etc., hasbeen known. As shown in FIGS. 16, 17A and 17B, this linear guideapparatus comprises a guide rail 2 having a rolling-element rollinggroove 1 formed on both sides thereof in the axial direction and aslider 3 which fits in the guide rail 2. Provided on the inner surfaceof the slider 3 is a rolling-element rolling groove 4 disposed opposedto the rolling-element rolling groove 1 of the guide rail 2. A number ofrolling elements 5 are rollably interposed between the rolling-elementrolling grooves 1, 4. In this arrangement, the slider 3 makes relativemovement with the rolling elements 5 along the axis of the guide rail 2.

In this linear guide apparatus, the slider 3 has an end cap 6 providedon both axial ends thereof. The end cap 6 has a circulating path 7forming a curved path communicating to the rolling-element rollinggrooves 1, 4 for circulating rolling elements.

As shown in FIG. 18, the slider 3 may have a slow inclination calledcrowning portion 8 provided at both ends thereof for relaxing stressconcentration. It is known that the provision of the crowning portion 8makes it possible to enhance durability. On the other hand, the crowningportion 8 has an effect on the degree of change of sliding resistance ofthe slider 3 of the linear guide apparatus during movement. This changeof sliding resistance has an adverse effect on the performance ofmachines comprising the linear guide apparatus. For example, thepositioning properties of the machines are deteriorated to disadvantage.It has thus been desired to reduce the change of sliding resistance.

However, the selection of the crowning form for inhibiting the change ofsliding resistance has heretofore been often experimentally made.Therefore, in order to realize a rolling linear guide having a reducedchange of sliding resistance, production on a trial basis and byexperiments must be repeated, requiring much labor.

SUMMARY OF THE INVENTION

The present invention has been worked out under these circumstances. Anaim of the present invention is to provide a linear guide apparatuswhich has a reduced change of an axial component of contact load of aslider at a crowning portion with movement of rolling elements toexhibit a reduced change of sliding resistance.

Another aim of the present invention is to provide a linear guideapparatus suitable for uses in situations requiring a reduced change ofsliding resistance such as in an electric discharge machine, a moldprocessing machine, a drawing device, a semiconductor producing machine(exposing device), and a precision measuring instrument.

In order to accomplish the aforementioned aims of the present invention,the present invention lies in a linear guide apparatus having a guiderail having a rolling-element rolling groove formed on a side thereofalong an axial direction, and a slider having a rolling-element rollinggroove disposed so as to be opposed to the rolling-element rollinggroove of the guide rail, the slider moving relatively in the axialdirection of the guide rail with rolling elements rollably interposedbetween the two rolling-element rolling grooves, and a crowning portiondisposed on both axial ends of the rolling-element rolling groove of theslider, wherein a maximum value of a total load in an axial directionapplied on the crowing portions on the axial ends of the slider by therolling elements is set to be smaller than a maximum value of load inthe axial direction applied on one of the crowning portions of theslider by one rolling element.

In other words, the linear guide apparatus is arranged such that theload in the axial direction applied on the rolling elements at the rightand left crowning portions cancel each other. In this arrangement, thesum of the load applied on the rolling elements at the right and leftcrowning portions is set to be smaller than the maximum value of theload in the axial direction applied on the slider at one of the crowningportions by the rolling elements.

At the portion free of crowning portion, the direction of contact loadof the rolling elements with the rolling groove is perpendicular to theaxial direction. Therefore, the contact load at this portion has noaxial component.

However, at the portion having a crowning portion, the contact load ofthe rolling element with the rolling groove has an axial component. Themagnitude of this axial component normally changes with the movement ofthe rolling elements. This is attributed to the fact that thearrangement of the crowning portion causes the change of the rollingelements and the rolling groove and hence the change of the contact loadand the direction of contact of the rolling elements with the rollinggroove changes.

The change of the axial load generated at the crowning portion causesthe sum of the load in the axial direction applied on the slider by thecrowning portion at both ends to change with the movement of the slider(i.e., movement of the rolling elements). This change of the load in theaxial direction directly leads to the change of the sliding resistanceof the slider.

In accordance with the present invention, these axial loads can canceleach other at the right and left crowning portions. In this arrangement,the axial load acted on the slider can be predetermined to be smallerthan the maximum value of the load in the axial direction applied on theslider at one of the crowning portions. In other words, the change ofthe sliding resistance of the slider can be reduced.

In addition to this, according to the present invention, it ispreferable that a linear guide apparatus having a guide rail having arolling-element rolling groove formed on a side thereof along in anaxial direction, a slider having a rolling-element rolling groovedisposed so as to be opposed to the rolling-element rolling groove ofthe guide rail, the slider moving relatively in the axial direction ofthe guide rail with rolling elements rollably interposed between the tworolling-element rolling grooves, and a crowning portion disposed on theboth axial ends of the slider, wherein the crowning portion is a linearcrowning having a constant inclination angle θ and satisfying thefollowing relationship:

0<θ(N×t−Le)<δ_(o)

wherein N represents an integer of 1 or more; t represents the distancebetween centers of the rolling elements (with separator) or a diameterof the rolling element (free of separator); Le represents the length ofthe non-crowning portion; and δ_(o), represents preload amount (diameterof rolling element based on the diameter of rolling element which giveszero elastic deformation at the non-crowning portion).

Further, according to the present invention, it is preferable that alinear guide apparatus having a guide rail having a rolling-elementrolling groove formed on a side thereof along in an axial direction, anda slider having a rolling-element rolling groove disposed so as to beopposed to the rolling-element rolling groove of the guide rail, theslider moving relatively in the axial direction of the guide rail withrolling elements rollably interposed between the two rolling-elementrolling grooves, and a crowning portion disposed on the both axial endsof the slider, wherein the crowning portion is an arc crowning having aconstant radius R and satisfying any of the following relationships (1)to (3):

Nt−Le≦0.5t and 0.5{square root over (2Rδ _(o))}<N×t−Le<1.5{square rootover (2Rδ _(o))};  (1)

0.5t<N×t−Le <1.5{square root over (2Rδ _(o))};and  (2)

t<{square root over (2Rδ _(o))}  (3)

wherein N represents an integer of 1 or more; t represents the distancebetween centers of the rolling elements (with separator) or the diameterof the rolling element (free of separator); Le represents the length ofthe non-crowning portion; and δ_(o) represents preload amount (diameterof rolling element based on the diameter of rolling element which giveszero elastic deformation at the non-crowning portion).

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a longitudinal section of a linear crowning according to afirst embodiment of the present invention;

FIG. 2 is a diagram of the first embodiment of the present inventionillustrating the state that the center of rolling element L1 ispositioned at the left crowning starting point;

FIG. 3 is a diagram of the first embodiment of the present inventionwherein a separator is disposed between the rolling elements;

FIG. 4 is a diagram of the first embodiment of the present inventionillustrating the state that the various rolling elements have movedleftward by ξ_(L1);

FIG. 5 is a diagram of the first embodiment of the present inventionillustrating the axial load applied on the slider;

FIG. 6 is a diagram of the first embodiment of the present inventionillustrating the axial load applied on the slider;

FIG. 7 is a diagram of the first embodiment of the present inventionillustrating the change of Fsmax/Fmax with L_(CL);

FIG. 8 is a diagram of the first embodiment of the present inventionillustrating the change of Fsmax/Fmax with L_(CL);

FIG. 9 is a graph of the first embodiment of the present inventionillustrating the results of the change of sliding resistance with themovement of the slider;

FIG. 10 is a longitudinal section of a second embodiment of the presentinvention illustrating an arc crowning;

FIG. 11 is a diagram of the second embodiment of the present inventionillustrating the axial load applied on the slider;

FIG. 12 is a diagram of the second embodiment of the present inventionillustrating the axial load applied on the slider;

FIG. 13 is a diagram of the second embodiment of the present inventionillustrating the change of Fsmax/Fmax with L_(CR);

FIG. 14 is a diagram of the second embodiment of the present inventionillustrating the change of Fsmax/Fmax with L_(CR);

FIG. 15 is a graph of the second embodiment of the present inventionillustrating the results of the change of sliding resistance with themovement of the slider;

FIG. 16 is a partially cut away perspective view of a related art linearguide apparatus;

FIGS. 17A and 17B illustrate linear guide apparatus wherein FIG. 17A isa transverse sectional view of the linear guide apparatus and FIG. 17Bis an enlarged view of part A of FIG. 17A; and

FIG. 18 is a diagram illustrating a crowning portion of the linear guideapparatus of FIGS. 17A and 17B.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Embodiments of implementation of the present invention will be describedhereinafter in connection with the attached drawings.

FIG. 1 illustrates a first embodiment of the present invention. Where inthe parts are the same as those of the related art linear guideapparatus of FIGS. 16 to 18, the same numerals are used and theirdescription will be omitted. As shown in FIG. 1, a slider 3 which makesrelative movement along the axis of a guide rail 2 with rolling elements5 has a linear crowning 11 disposed at both ends thereof.

Both the right and left linear crownings 11 have the same shape and aconstant inclination angle θ. The rolling element 5 acts as a ballcrowning. Supposing that the distance in the axial direction from thecrowning starting point a over the rolling element 5 of the linearcrowning 11 is ξ, the following equation is established.

y=θξ

(y: Crowning in the direction of contact of rolling element)$Q = {{k_{n}\left( {\delta_{0} - {\theta \quad \xi}} \right)}^{1.5} = {Q_{0}\left( {1 - \frac{\xi}{L_{CL}}} \right)}^{1.5}}$

(Q: contact load)$F = {{{Q\quad \sin \quad \theta} \approx {Q\quad \theta}} = {Q_{0}\quad {\theta \left( {1 - \frac{\xi}{L_{CL}}} \right)}^{1.5}}}$

(F: magnitude of axial component of contact load)

wherein θ: Angle of inclination of crowning

δ₀: Preload amount of rolling element at portion free of crowning onrace when unloaded (diameter of rolling element based on the diameter ofrolling element which gives zero elastic deformation)

k_(n): Constant determined by Hertz's theory

L_(CL)=δ₀/θ

Q₀=k_(n)δ₀ ^(1.5)

Among the rolling elements 5 disposed in the linear crowning 11, onlythe rolling elements 5 satisfying the relationship ξ≦L_(CL) are given aload. Let us firstly suppose the case where the maximum number of therolling elements 5 given a load at the linear crowing 11 is 1, i.e.,L_(CL)≦t.

Let us then suppose the state that the center of the rolling element Liis positioned at the left crowning starting point a as shown in FIG. 2.The rolling elements 5 juxtaposed in the rolling-element rolling grooves1 and 4 are all disposed in a line at a constant distance t between thecenters thereof. The symbol t indicates the distance between the centerof the rolling elements 5 with a separator 12, if any, interposedbetween the rolling elements (as shown in FIG. 3).

In the case where there is no separator 12 disposed between the rollingelements 5 (as shown in FIG. 2), t is the same as the diameter Da of therolling element 5. Examples of the separator 12 include a gathering ofseparated separators 12 and a series combination of separators 12.Either of these separators 12 may be used. These separators 12 mayoccasionally have no rolling element 5 present on some portion of theends thereof. However, this portion free of rolling element accounts fora small proportion in the total area and thus can be neglected.

In the state shown in FIG. 3, the rolling element 5 disposed leftmostamong the rolling elements 5 in the left crowning is designated as R1.The distance S from the center of the rolling element R1 to the rightcrowning starting point a is calculated by the following equation:

S=N×t−Le

(N: integer selected satisfying 0≦S<t)

Let us next suppose the state that the various rolling elements 5 eachmove leftward by ξ_(L1) (as shown in FIG. 4). The disposition of therolling elements 5 occurring when ξ_(L1) rises to t is the same as thatoccurring when ξ_(L1) is zero. Accordingly, the range of ξ_(L1) may beconsidered to be 0≦ξ_(L1)≦t.

When ξ_(L1) satisfies the relationship 0≦ξ_(L1)≦t, three rollingelements L1, R1 and R2 give an axial load to the slider 3. The axialloads F_(L1), F_(R1) and F_(R2) applied on the slider 3 by the rollingelements 5 are represented by the following equations with the provisothat the rightward load is positive:$F_{L\quad 1} = {Q_{0}\quad {\theta \left( {1 - \frac{\xi_{L\quad 1}}{L_{CL}}} \right)}^{1.5}}$

(F_(L1) is zero when ξ_(L1) doesn't satisfy the relationship0≦ξ_(L1)≦L_(CL))$F_{R\quad 1} = {{- Q_{0}}\quad {\theta \left( {1 - \frac{S - \xi_{L1}}{L_{CL}}} \right)}^{1.5}}$

(F_(R1) is zero when ξ_(L1) doesn't satisfy the relationshipS−L_(CL)≦ξ_(L1)≦S)$F_{R2} = {{- Q_{0}}\quad {\theta \left( {1 - \frac{S + t - \xi_{L1}}{1}} \right)}^{1.5}}$

(F_(R2) is zero when ξ_(L1) doesn't satisfy the relationshipS+t−L_(CL)≦ξ_(L1)≦S+t)

The sum Fs of the axial loads applied on the slider 3 is represented bythe following equation:

Fs=F _(L1) +F _(R1) +F _(R2)

An example of the relationship between ξ_(L1) and F_(L1), F_(R1), F_(R2)and Fs is shown in FIG. 5. As ξ_(L1) changes, Fs changes. In otherwords, as the rolling elements 5 move, the axial load given by thecrowning portion changes. This change directly leads to the change ofsliding resistance. The maximum value of the axial load applied on theslider 3 by one of the rolling elements 5 at the crowning portion isdefined to be Fmax. Fmax is Q₀θ.

Let us next suppose the case where the maximum number of the rollingelements 5 given a load at the crowning portion is 2 or more (L_(CL)>t).Let us suppose the case where t<L_(CL)≦2t by way of example. When ξ_(L1)satisfies the relationship 0≦ξ_(L1)≦t, five rolling elements L₁, L₂, R₁,R₂ and R₃ give an axial load to the slider 3. The axial loads F_(L1),F_(L2), F_(R1), F_(R2) and F_(R3) applied on the slider by the rollingelements are shown in FIG. 6. FIG. 6 shows the sum Fs of the axial loadsapplied on the slider.

FIGS. 7 and 8 each illustrate the change of Fsmax/Fmax with L_(CL). FIG.7 illustrates the case where 0<S<0.370 t. FIG. 8 illustrates the casewhere 0.370 t≦S<t. By selecting proper L_(CL), the relationshipFsmax<Fmax can be established. This is because when L_(CL) is proper,the axial load given at the right and left crowning portions cancel eachother.

When S is zero, the relationship Fsmax=Fmax can be establishedregardless of L_(CL). FIGS. 7 and 8 each illustrate only the case where0≦L_(CL)≦2t. However, even when L_(CL) falls outside this range, if S isnot zero, the relationship Fsmax<Fmax can be established at L_(CL) ofnot smaller than t regardless of L_(CL).

As can be seen in FIGS. 7 and 8, the relationship Fsmax<Fmax can beestablished if the following relationship is satisfied. In other words,the maximum value of the axial load applied on the slider 3 can bepredetermined to be smaller than the maximum value of the axial loadapplied on the crowning portion by one of the rolling elements.

0<S<L_(CL)  (1)

Further, in the case where 0<S<0.370t, Fsmax/Fmax can be predeterminedminimum when L_(CL) is 2.70S.

The relationship (1) and the definition of S and L_(CL) make it possibleto establish the relationship Fsmax<Fmax if there is an integer Nsatisfying the following relationship:

0<θ(N×t−Le)<δ₀  (2)

The preload amount δ₀ varies with the properties (rigidity, magnitude ofsliding resistance, etc.) required for rolling linear guide apparatus.It is preferred that the aforementioned relationship be satisfied evenfor different δ₀ values by the use of a slider having the same crowningshape. This is because the number of kinds of sliders to be designed andproduced can be reduced.

Among uses requiring that the variation of sliding resistance be small,uses subject to varying load such as mold processing machine requiresome rigidity. In such uses, the preload amount δ₀ in the linear guideapparatus is normally not smaller than 0.00002 Da. As can be seen in therelationship (2), when there is an integer N satisfying therelationship, the relationship Fsmax<Fmax is established if δ₀ is notgreater than 0.0002 Da.

0<θ(N×t−Le)<0.0002Da  (3)

On the other hand, uses requiring accurate position control such assemiconductor producing machine (exposing device) and precisionmeasuring instrument require that the variation as well as the absolutevalue of sliding resistance be small. In such uses, the preload amountδ₀ in the linear guide apparatus is normally not smaller than 0.0001 Da.As can be seen in the relationship (2), when there is an integer Nsatisfying the relationship, the relationship Fsmax<Fmax is establishedwith any δ₀ of not greater than 0.0001 Da.

0<θ(N×t−Le)<0.0001Da  (4)

In order to reduce the variation of sliding resistance in particular,Fsmax/Fmax may be predetermined to be minimum. As can be seen in FIG. 7,the conditions under which Fsmax/Fmax is minimum (0.5) give thefollowing relationships:

 0<N×t−Le<0.370t and θ(N×t−Le)=0.370δ₀  (5)

It is normally difficult to strictly satisfy the relationship (5) due torestrictions such as error in processing. Let us then examine theconditions under which Fsmax/Fmax is smaller than e. These conditionsgive the following relationships:

0<N×t−Le<0.370t and βe _(L)δ₀<θ(N×t−Le)γe _(L)δ₀  (6)

wherein βe_(L) and γe_(L) each are a value determined by e. Table 1shows βe_(L) and γe_(L) values for various e values. For example, when eis 80%, βe_(L) and γe_(L) are 0.138 and 0.658, respectively, as setforth in Table 1. Accordingly, when the following relationships aresatisfied, the relationship Fsmax/Fmax<80% can be established.

0<N×t−Le<0.370t and 0.138δ₀<θ(N×t−Le)<0.658δ₀  (6)

TABLE 1 e β_(eL) γ_(eL) 100% 0 1  90% 0.0678 0.785  80% 0.138 0.658  70%0.212 0.552  60% 0.289 0.457

Explaining next specific examples of the linear guide apparatus, theguide rail 2 has two lines of rolling-element rolling grooves 1 providedon both sides thereof and the slider 3 has a linear crowning 11 providedon both moving ends thereof. The rolling element 5 is a ball. Thecontact angle α is 50°.

EXAMPLE 1-1

Le satisfying the relationship Fsmax<Fmax for any δ₀ of not smaller than0.0001 Da

Ball diameter: Da=5.56 mm

Inclination of crowning: θ=0.001 rad

Distance between the center of adjacent rolling elements: t=5.56 mm(free of separator)

The relationship (4) can be deformed to the following relationship:${{N \times t} - \frac{0.0001{\quad \quad}D\quad a}{\theta}} < {L\quad e} < {N \times t}$

Substituting various values for the factors in the foregoingrelationship gives the following relationships for the range of Le:

5.00 mm<Le<5.56 mm (N=1)

10.6 mm<Le<11.1 mm (N=2)

43.9 mm<Le<44.5 mm (N=8)

Le is then selected from the aforementioned ranges (e.g., 44 mm). As aresult, the relationship Fsmax<Fmax can be established for any δ₀ of notsmaller than 0.0001 Da, i.e., not smaller than 0.556 μm.

EXAMPLE 1-2

t satisfying the relationship Fsmax<Fmax for any δ₀ of not smaller than0.0002 Da

Ball diameter: Da=5.56 mm

Inclination of crowning: θ=0.001 rad

Length of non-crowning portion: Le=46 mm

The relationship (3) can be deformed to the following relationship:$\frac{L\quad e}{N} < t < {\frac{1}{N}\left( {{L\quad e} + \frac{0.0002\quad D\quad a}{\theta}} \right)}$

wherein t is not smaller than Da. Substituting various values for thefactors in the foregoing relationship gives the following relationshipsfor the range of Le:

46.00 mm<t<47.1 mm (N=1)

23.0 mm<t<23.6 mm (N=2)

5.75 mm<t<5.89 mm (N=8)

The thickness of the separator 12 is then predetermined such that tfalls within the aforementioned range (e.g., t=5.8 mm). As a result, therelationship Fsmax<Fmax can be established for any δ₀ of not smallerthan 0.0002 Da, i.e., not smaller than 1.11 μm.

EXAMPLE 1-3

Case of Determination of Le

Ball diameter: Da=5.56 mm

Inclination of crowning: θ=0.001 rad

Distance between the center of adjacent rolling elements: t=5.56 mm(free of separator)

Preload amount: δ₀=0.004 mm

The conditions (2) under which the relationship Fsmax<Fmax isestablished can be deformed to the following relationships:

N×t−0.370t<Le<N×t and

${{N \times t} - \frac{\delta_{0}}{\theta}} < {L\quad e} < {N \times t}$

Substituting various values for the factors in the foregoingrelationship gives the following relationships:

3.50 mm<Le<5.56 mm

9.06 mm<Le<11.1 mm

42.4 mm<Le<44.5 mm

48.0 mm<Le<50.0 mm

By properly selecting Le from the aforementioned ranges, therelationship Fsmax<Fmax can be established.

Let us next consider the range within which the relationshipFsmax/Fmax<70% is established. The relationship (6) can be deformed tothe following relationship:

N×t−0.370t<Le<N×t and

${{N \times t} - \frac{\gamma_{e\quad L}\delta_{0}}{\theta}} < {L\quad e} < {{N \times t} - \frac{\beta_{e\quad L}\delta_{0}}{\theta}}$

When e is 70%, β_(eL) and γ_(eL) are 0.212 and 0.552, respectively, asset forth in Table 1. Substituting various values for the factors in theforegoing relationships gives the following relationships for the rangeof Le. The Le value (Les) which gives minimum Fsmax/Fmax can bedetermined by the relationship (5).

3.50 mm<Le<4.71 mm, Les=4.08 mm (N=1)

9.06 mm<Le<10.3 mm, Les=9.64 mm (N=2)

42.4 mm<Le<43.6 mm, Les=43.0 mm (N=3)

48.0 mm<Le<49.2 mm, Les=48.6 mm (N=4)

By properly selecting Le from the aforementioned ranges, therelationship Fsmax/Fmax<70% can be established.

The closer to Les Le is, the smaller can be Fsmax/Fmax.

The results (numerical analysis) of examination of change of slidingresistance with the movement of the slider 3 for Le of 43 mm, 43.5 mmand 45 mm are shown in FIG. 9. In FIG. 9, the abscissa indicates themoving distance of the slider 3 and the ordinate indicates the slidingresistance of the slider 3.

When Le is 45 mm, the relationship Fsmax<Fmax cannot be satisfied.

When Le is 43 mm or 43.5 mm, the relationship Fsmax/Fmax<70% can besatisfied.

The Le value of 43 mm gives minimum Fsmax/Fmax. Therefore, the variationof sliding resistance is particularly small.

As mentioned above, the advantage of the present invention can beconfirmed.

EXAMPLE 1-4

Case of Determination of t

The preload amount in the linear guide apparatus is variouslypredetermined depending on the working conditions. In Example 1-3, inorder to secure similar Fsmax/Fmax when the preload amount varies, it isnecessary that the length of the non-crowning portion Le be changed.However, this approach is undesirable from the standpoint of productionand stock control. Therefore, it is arranged such that the distance tbetween the center of the adjacent rolling elements varies with preloadamount by disposing a separator 12 interposed between the rollingelements 5. The separator 12 can be easily mass-produced by injectionmolding or like methods. Even separators 12 having different thicknessescan be easily controlled in production and stock.

Ball diameter: Da=5.56 mm

Inclination of crowning: θ=0.0001 rad

Length of non-crowning portion: Le=46 mm

Preload amount: δd=0.004 mm

Let us next consider the conditions under which the relationshipFsmax/Fmax<80% is established. The relationship (6) can be deformed tothe following relationships:$\frac{L\quad e}{N} < t < {\frac{L\quad e}{N - 0.370}\quad {and}\quad \frac{1}{N}\left( {\frac{\beta_{e\quad L}\delta_{0}}{\theta} + {L\quad e}} \right)} < t < {\frac{1}{N}\left( {\frac{\gamma_{e\quad L}\delta_{0}}{\theta} + {L\quad e}} \right)}$

When e is 80%, β_(eL) and γ_(eL) are 0.138 and 0.658, respectively, asset forth in Table 1. Le us next substitute various values for thefactors in these relationships. Since t is not smaller than Da, thefollowing relationships can be established for the range of t. The valueof t (ts) which gives minimum Fsmax/Fmax can be determined by therelationship (5).

46.6 mm<t<48.6 mm, ts=47.5 mm (N=1)

23.3 mm<t<24.3 mm, ts=23.7 mm (N=2)

5.82 mm<t<6.03 mm, ts=5.93 mm (N=8)

The value of t is selected from the aforementioned ranges. The value oft is preferably as small as possible. This is because when the number ofrolling elements in the rolling-element rolling grooves 1 and 4increases, the load capacity increases. Further, the closer to ts thevalue of t is, the smaller can be Fsmax/Fmax. For example, when t is 5.9mm, Fsmax/Fmax can be predetermined to be smaller than 80%.

Let us next suppose that the preload amount δ₀ is 0.001 mm. In thiscase, too, the following relationships are established for the range oft and the value of ts:

46.1 mm<t<46.7 mm, ts=46.4 mm (N=1)

23.1 mm<t<23.3 mm, ts=23.2 mm (N=2)

 5.76<t<5.83 mm, ts=5.8 mm (N=8)

The value of t is then selected from the aforementioned ranges. Forexample, when t is 5.8 mm, Fsmax/Fmax can be predetermined to be smallerthan 80%.

As mentioned above, when separators 12 having different dimensions areused, similar Fsmax/Fmax values can be obtained even if differentpreload amounts are used with the same slider 3.

EXAMPLE 1-5

Case where the Range of Preload Amount is Known Le at which theRelationship Fsmax/Fmax<80% is Secured

In Examples 1-3 and 1-4, sliders 3 or separators 12 having differentdimensions were required. However, when the range of preload amount ispreviously known, a slider 3 and a separator 12 having the samedimension can be used to predetermine Fsmax/Fmax to be within a requiredrange. This is advantageous in production and stock control.

Ball diameter: Da=5.56 mm

Inclination of crowning: θ=0.001 rad

Length of non-crowning portion: Le=46 mm

Minimum preload amount: δmin=0.002 mm

Maximum preload amount: δmax=0.008 mm

As can be seen in the relationship (6), when the following relationshipsare satisfied, Fsmax can be predetermined to be smaller than e for δ₀ offrom not smaller than δmin to not greater than δmax.$\frac{L\quad e}{N} < t < {\frac{L\quad e}{N - 0.370}\quad {and}\quad \frac{1}{N}\left( {\frac{\beta_{e\quad L}\delta_{\max}}{\theta} + {L\quad e}} \right)} < t < {\frac{1}{N}\left( {\frac{\gamma_{e\quad L}\delta_{\min}}{\theta} + {L\quad e}} \right)}$

When e is 80%, β_(eL) and γ_(eL) are 0.138 and 0.658, respectively, asset forth in Table 1. Let us next substitute various values for thefactors in the aforementioned relationships. As a result, the followingrelationships are established for the range of t:

47.1 mm<t<47.3 mm (N=1)

23.6 mm<t<23.7 mm (N=2)

5.89 mm<t<5.91 mm (N:=8)

The value of t is then selected from the aforementioned ranges. Forexample, t is determined to be 5.9 mm. Then, Fsmax/Fmax can bepredetermined to be smaller than 80%. As mentioned above, even when aslider 3 and a separator having the same dimension are used anddifferent preload amounts are used, Fsmax/Fmax can be predetermined tobe within the same range.

In the case where β_(eL)δmax is not smaller than γ_(eL)δmin, when thepreload amount falls within a required range, Fsmax/Fmax cannot bepredetermined to be smaller than e with the same value of t. In thiscase, it is necessary that a plurality of kinds of separators havingdifferent values of t be prepared.

FIG. 10 illustrates a second embodiment of the present invention whichis an arc crowning 13. The arc crowning 13 comprises a right portion anda left portion having the same shape and a constant radius R. Therolling element 5 is a ball. Supposing that the distance of the rollingelement 5 in the arc crowning 13 from the crowning starting point a isξ, the following relationships are established:$y = \frac{\xi^{2}}{2\quad R}$

(y: crowning)$Q = {{k\quad {n\left( {\delta_{0} - \frac{\xi^{2}}{2R}} \right)}^{1.5}} = {Q_{0}\left\{ {1 - \left( \frac{\xi}{L_{CR}} \right)^{2}} \right\}^{1.5}}}$

(Q: contact load)$F = {{{Q\quad \sin \quad \theta} \approx {Q\quad \frac{y}{\xi}}} = {Q_{0}\quad \frac{\xi}{R}\left\{ {1 - \left( \frac{\xi}{L_{CL}} \right)^{2}} \right\}^{1.5}}}$

(F: magnitude of axial component of contact load)

wherein R: Radius of crowning portion

δ₀: Elastic deformation of ball at crowning-free portion on race whenunloaded

k_(n): Constant determined by Hertz's theory

L_(CR)={square root over (2Rδ₀)}

Q₀=k_(n)2δ₀ ^(1.5)

Let us consider hereinafter the arc crowning 13 similarly to theaforementioned linear crowning 11. Let us firstly suppose the case wherethe number of rolling elements given a load at the crowning portion is 1or less, i.e., L_(CR)≦t (FIG. 2). Let us consider 0≦ξ_(L1)≦t. Therolling elements giving an axial load to the slider 3 are rollingelements L1, R1 and R2. Supposing that the distance of the rollingelement L1 from the left crowning starting point is ξ_(L1), the axialload Fs applied on the slider 3 is represented by the followingequation:

Fs=F _(L1) +F _(R1) +F _(R2)

F_(L1), F_(R1) and F_(R2) are represented by the following equations:$\begin{matrix}\begin{matrix}{F_{L1} = {Q_{0}\frac{\xi_{L1}}{R}\left\{ {1 - \left( \frac{\xi_{L1}}{L_{CR}} \right)^{2}} \right\}^{1.5}}} \\{F_{R1} = {{- Q_{0}}\frac{D - \xi_{L1}}{R}\left\{ {1 - \left( \frac{D - \xi_{L1}}{L_{CR}} \right)^{2}} \right\}^{1.5}}}\end{matrix} \\{F_{R2} = {Q_{0}\frac{D + t_{w} - \xi_{L1}}{R}\left\{ {1 - \left( \frac{D + t_{w} - \xi_{L1}}{L_{CR}} \right)^{2}} \right\}^{1.5}}}\end{matrix}$

The relationship between ξ_(L1) and F_(L1), F_(R1), F_(R2) and Fs isshown in FIG. 11. The maximum value of the axial load applied on theslider 3 by one of the rolling elements in the crowning portion isdefined to be Fmax. Fmax is represented by the following equation:$F\quad \max \times \frac{3\sqrt{3}Q_{0}L_{CR}}{16\quad R}$

Let us next consider the case where there are two or more rollingelements 5 given a load at the crowning portion (L_(CR)>t). Let usconsider the case of t<L_(CR)≦2 t by way of example. When ξ_(L1) fallswithin the range of from not smaller than 0 to not greater than t, thefive rolling elements L1, L2, R1, R2 and R3 give a load in the axialdirection. The axial loads F_(L1), F_(L2), F_(R1), F_(R2) and F_(R3)applied on the slider 3 by the rolling elements 5 are shown in FIG. 12.FIG. 12 indicates the sum Fs of axial loads applied on the slider 3.

FIGS. 13 and 14 each illustrate the change of Fsmax/Fmax with L_(CR).FIG. 13 illustrates the case of 0≦S≦0.5t. FIG. 14 illustrates the caseof 0.5t≦S≦t. FIGS. 13 and 14 each illustrate only the case where0≦L^(CR)≦2t. However, even when L_(CR) falls outside this range, therelationship Fsmax≦Fmax can be established at L_(CR) of not smaller thant regardless of L_(CR).

As shown in FIGS. 13 and 14, when any of the clauses (1) to (3) of therelationship (11) is satisfied, Fsmax can be predetermined to be smallerthan Fmax. In other words, the maximum value of the axial load acted onthe slider 3 can be predetermined to be smaller than the maximum valueof the axial load applied on the slider by one rolling element at thecrowning portion. $\begin{matrix}{S \leq {0.5\quad t\quad {and}{\quad \quad}\frac{2}{3}S} < L_{CR} < {2\quad S}} & (1) \\{S > {0.5\quad t\quad {and}\quad \frac{2}{3}S} < L_{CR}} & (2)\end{matrix}$

 (3) L _(CR) <t  (11)

Further, when the following relationships are satisfied, Fsmax/Fmaxbecomes minimum.

The minimum value is substantially zero.

L _(CR)=1.02(S+M×t)

(M: integer of not smaller than 0)

Accordingly, when any of the clauses (1) to (3) of the followingrelationship (12) is satisfied, Fsmax can be predetermined to be smallerthan Fmax.

(1) N×t−Le≦0.5t and 0.5{square root over (2Rδ ₀)}<N×t−Le<1.5{square rootover (2Rδ ₀)}

(2) 0.5t<N×t−Le<1.5{square root over (2Rδ ₀)}

(3) t<{square root over (2Rδ₀)}  (12)

The following relationships are established as in the case of linearcrowning.

Conditions under which Fsmax<Fmax is established in uses requiring somerigidity (0.0002 Da<δ₀):

0.5t<N×t−Le <1.5{square root over (0.0004RDa)} or t<{square root over(0.0004 RDa)}  (13)

Conditions under which Fsmax<Fmax is established in uses requiring asmall sliding resistance (0.0001 Da<δ₀):

0.5t<N×t−Le <1.5{square root over (0.0002RDa)} or t<{square root over(0.0002 RDa)}  (14)

Conditions under which Fsmax/Fmax is minimum:

N×t−Le=0.982{square root over (2Rδ ₀)}  (15)

Conditions under which Fsmax/Fmax is smaller than e:

β_(eR){square root over (2Rδ ₀)}<N×t−Le<γ _(eR){square root over (2Rδ₀)}  (16)

The relationship between e and β_(eR), γ_(eR) is set forth in Table 2.For example, when e is 80%, the following relationship is established:

0.693{square root over (2Rδ ₀)}<N×t−Le<1.29{square root over (2Rδ ₀)}

TABLE 2 e β_(eL) γ_(eL) 100% 0.5 1.5  90% 0.635 1.36  80% 0.693 1.29 70% 0.737 1.24  60% 0.776 1.20  50% 0.811 1.15  40% 0.844 1.12  30%0.877 1.08  20% 0.910 1.04  10% 0.946 1.01

Explaining next specific examples of the linear guide apparatus, theguide rail 2 has two lines of rolling-element rolling grooves 1 providedon both sides thereof and the slider 3 has an arc crowning 13 providedon both moving ends thereof. The rolling element 5 is a ball. Thecontact angle α is 50°.

EXAMPLE 2-1

Le satisfying the relationship Fsmax<Fmax for any δ₀ of not smaller than0.0002 Da

Ball diameter: Da=5.56 mm

Radius of crowning: R=2,000 mm

Distance between the center of adjacent rolling elements: t=5.56 mm(free of separator)

The relationship (13) can be deformed to the following relationship:

N×t−1.5{square root over (0.0004RDa)}<Le<(N−0.5)×t

Substituting various values for the factors in the foregoingrelationship gives the following relationships for the range of Le:

2.40 mm<Le<2.78 mm (N=1)

7.96 mm<Le<8.34 mm (N=2)

46.9 mm<Le<47.3 mm (N=9)

Le is then selected from the aforementioned ranges (e.g., 47 mm). As aresult, the relationship Fsmax<Fmax can be established for any δ₀ of notsmaller than 0.002 Da, i.e., not smaller than 1.11 μm.

EXAMPLE 2-2

Case of Determination of t

Ball diameter: Da=5.56 mm

Radius of crowning: R=2,000 mm

Length of non-crowning portion: Le=44 mm

Preload amount: δ₀=0.004 mm

The conditions (12) under which the relationship Fsmax<Fmax isestablished can be deformed to the following relationships:${\frac{1}{N}\left( {{0.5\sqrt{2R\quad \delta_{0}}} + {L\quad e}} \right)} < t < {\frac{1}{N}\left( {{1.5\sqrt{2R\quad \delta_{0}}} + {L\quad e}} \right)}$

Substituting various values for the factors in the aforementionedrelationship gives the following relationships because t is smaller thanDa:

46.0 mm<t<50.0 mm (N=1)

23.0 mm<t<25.0 mm (N=2)

 6.75 mm<t<7.14 mm (N=7)

5.75 mm<t<6.25 mm (N=8)

By properly selecting t from the aforementioned ranges, the relationshipFsmax<Fmax can be established. Let us next consider the range withinwhich the relationship Fsmax/Fmax<60% is established. The relationship(16) can be deformed to the following relationship:${\frac{1}{N}\left( {{\beta_{e\quad R}\sqrt{2R\quad \delta_{0}}} + {L\quad e}} \right)} < t < {\frac{1}{N}\left( {{\gamma_{e\quad R}\sqrt{2R\quad \delta_{0}}} + {L\quad e}} \right)}$

When e is 60%, β_(eR) and γ_(eR) are 0.776 and 1.20, respectively, asset forth in Table 2. Substituting various values for the factors in theforegoing relationships gives the following relationships for the rangeof t. The Le value (ts) which gives minimum Fsmax/Fmax can be determinedby the relationship (15).

47.1 mm<t<48.8 mm, ts=47.9 mm (N=1)

23.6 mm<t<24.4 mm, ts=24.0 mm (N=2)

6.73 mm<t<6.97 mm, ts=6.85 mm (N=7)

5.89 mm<t<6.10 mm, ts=5.99 mm (N=8)

By properly selecting t from the aforementioned ranges, the relationshipFsmax/Fmax<60% can be established. The closer to ts t is, thesmaller-can be Fsmax/Fmax.

The results (numerical analysis) of examination of change of slidingresistance with the movement of the slider 3 for t of 6.0 mm, 6.1 mm and6.4 mm are shown in FIG. 15. In FIG. 15, the abscissa indicates themoving distance of the slider 3 and the ordinate indicates the slidingresistance of the slider 3.

When t is 6.4 mm, the relationship Fsmax<Fmax cannot be satisfied. Thevariation of sliding resistance is great.

When t is 6.0 mm or 6.1 mm, the relationship Fsmax/Fmax<60% can besatisfied. The variation of sliding resistance is great.

The t value of 6.0 mm is close to 5.99 mm, which value gives minimumFsmax/Fmax. Therefore, the variation of sliding resistance isparticularly small.

As mentioned above, the advantage of the present invention can beconfirmed.

EXAMPLE 2-3

Case where the Range of Preload Amount is Known The relationshipFsmax/Fmax<80% is then secured.

Ball diameter: Da=5.56 mm

Radius of crowing: R=1,000 mm

Length of non-crowning portion: Le=46 mm

Minimum preload amount: δmin=0.002 mm

Maximum preload amount: δmax=0.006 mm

As can be seen in the relationship (16), when the following relationshipis satisfied, Fsmax/Fmax can be predetermined to be smaller than e forδ₀ of from not smaller than δmin to not greater than δmax.${\frac{1}{N}\left( {{\beta_{e\quad R}\sqrt{2R\quad \delta_{\max}}} + {L\quad e}} \right)} < t < {\frac{1}{N}\left( {{\gamma_{e\quad R}\sqrt{2R\quad \delta_{\min}}} + {L\quad e}} \right)}$

When e is 80%, β_(eL) and γ_(eL) are 0.693 and 1.29, respectively, asset forth in Table 2. Let us next substitute various values for thefactors in the aforementioned relationship. As a result, the followingrelationships are established for the range of t:

46.5 mm<t<47.3 mm (N=1)

23.2 mm<t<23.7 mm (N=2)

5.81 mm<t<5.91 mm (N=8)

The value of t is then selected from the aforementioned ranges. Forexample, t is determined to be 5.9 mm. Then, Fsmax/Fmax can bepredetermined to be smaller than 80%.

In the foregoing embodiments, as preload amount there was used theelastic deformation of rolling element unloaded. In order to furtherreduce the variation of sliding resistance, the elastic deformation δ₁of rolling element under the conditions that the reduction of variationof sliding resistance is important may be used instead of δ₀. Forexample, in the case where the present invention is applied to machineson which a great external load is not acted such as semiconductorproducing device, the elastic deformation δ₁ of rolling element by itsown weight may be used.

The aforementioned embodiments have been described with reference to thelinear crowning 11 and the arc crowning 13. The conditions under whichthe relationship Fsmax<Fmax is established can be similarly found forother crowning forms.

In the case where crowning is represented by a function (y=f(ξ)) ofdistance ξ from the crowning starting point, the following equations canbe established: $\begin{matrix}{Q = {{k_{n}\left( {\delta_{0} - {f(\xi)}} \right)}^{1.5} = {{Q_{0}\left( {1 - \frac{f(\xi)}{\delta_{0}}} \right)}^{1.5}\left( {Q\text{:}{contact}\quad {load}} \right)}}} & ({A1}) \\{F = {{{Q\quad \sin \quad \theta} \approx {Q\frac{y}{\xi}}} = {Q_{0}\frac{y}{\xi}\left( {1 - \frac{f(\xi)}{\delta_{0}}} \right)^{1.5}\quad \left( {F\text{:}{magnitude}\quad {of}\quad {axial}\quad {component}\quad {of}\quad {contact}\quad {load}} \right)}}} & ({A2})\end{matrix}$

wherein

δ₀: Elastic deformation of ball at crowning-free portion on race whenunloaded

k_(n): Constant determined by Hertz's theory

Q₀=k_(n)δ₀ ^(1.5)

As can be seen in the equation (A2), the magnitude of the axial loadapplied on one rolling element by the crowning portion can be obtainedas a function of position ξ of rolling element. Accordingly, ξ isplotted as abscissa and F as ordinate. From this graph can be determinedthe maximum value Fmax of axial load F applied on the slide by onerolling element.

Let us next suppose the case where the number of rolling elements givena load in the crowning portion is 1 or less, i.e., L_(CR)≦t, similarlyto the case of linear and arc crownings. The rolling elements giving anaxial load to the slider are rolling elements L1, R1 and R2. Supposingthat the distance of the rolling element L1 from the left crowningstarting point is ξ_(L1), the axial load Fs applied on the slider isrepresented by the following equation:

Fs=F _(L1) +F _(R1) +F _(R2)  (A3)

wherein F_(L1), F_(R1) and F_(R2) are respectively represented by thefollowing equations: $\begin{matrix}{F_{L1} = \left. {Q_{0}\frac{y}{\xi}} \middle| {}_{\xi = \xi_{L1}}\left( {1 - \frac{f\left( \xi_{L1} \right)}{\delta_{0}}} \right)^{1.5} \right.} & ({A4}) \\{F_{R1} = \left. {{- Q_{0}}\frac{y}{\xi}} \middle| {}_{\xi = {D - \xi_{L1}}}\left( {1 - \frac{f\left( {D - \xi_{L1}} \right)}{\delta_{0}}} \right)^{1.5} \right.} & ({A5}) \\{F_{R2} = \left. {{- Q_{0}}\frac{y}{\xi}} \middle| {}_{\xi = {D + t - \xi_{L1}}}\left( {1 - \frac{f\left( {D + t - \xi_{L1}} \right)}{\delta_{0}}} \right)^{1.5} \right.} & ({A6})\end{matrix}$

From the aforementioned equations (A3) to (A6) can be known therelationship between the coordinate ξ_(L1) of the rolling element L1 andthe sum Fs of axial loads applied on the slider. Accordingly, bygraphically plotting the relationship between ξ_(L1) and Fs, the maximumvalue Fsmax of the sum Fs of axial loads applied on the slider can beknown.

The present embodiment has been considered with reference to the case ofL_(CR)≦t. In the case of L_(CR)>t, too, Fmax and Fsmax can be determinedas already described in the case of linear and arc crownings.

As mentioned above, the maximum value Fmax of axial load applied on theslider by one rolling element and the maximum value Fsmax of the sum ofaxial loads applied on the slider by the rolling elements for the casewhere an arbitrary crowning form y=f(ξ) is given can be known.Accordingly, the crowning form y=f(ξ) giving Fmax>Fsmax can be found.

The present embodiment has been described only with reference to thecase where the rolling element is a ball. For the case where the rollingelement is a roller, too, the conditions under which Fsmax is smallerthan Fmax can be found.

In the case where the rolling element is a roller, the crowningrepresented by y=f(ξ) as a function of distance ξ from the crowningstarting point can be given the following equations: $\begin{matrix}{Q = {{k_{R}\left( {\delta_{0} - {f(\xi)}} \right)}^{1.11} = {{Q_{0}\left( {1 - \frac{f(\xi)}{\delta_{0}}} \right)}^{1.11}\left( {Q\text{:}{contact}\quad {load}} \right)}}} & ({A7}) \\{F = {{{Q\quad \sin \quad \theta} \approx {Q\frac{y}{\xi}}} = {Q_{0}\frac{y}{\xi}\left( {1 - \frac{f(\xi)}{\delta_{0}}} \right)^{1.11}\quad \left( {F\text{:}{magnitude}\quad {of}\quad {axial}\quad {component}\quad {of}\quad {contact}\quad {load}} \right)}}} & ({A8})\end{matrix}$

wherein

δ₀: Elastic deformation of ball at crowning-free portion on race whenunloaded

k_(n): Constant determined by Hertz's theory

Q₀=k_(n)δ₀ ^(1.11)

The foregoing equations (A7) and (A8) have the same form as theequations (A1) and (A2), in which the rolling element is a ball, exceptthat they have different exponents. Accordingly, the crowning formy=f(ξ) giving Fmax>Fsmax can be found in the same manner as in the casewhere the rolling element is a ball.

As mentioned above, in accordance with the present invention, the loadin the axial direction applied on the slider at the right and leftcrowning portions cancel each other. In this arrangement, the axial loadacted on the slider can be predetermined to be smaller than the maximumvalue of the axial load applied on the slider by one rolling element atone of the crowning portions. In other words, the change of the slidingresistance of the slider can be reduced.

What is claimed is:
 1. A linear guide apparatus comprising: a guide railhaving a rolling-element rolling groove formed on a side thereof alongin an axial direction; and a slider having a rolling-element rollinggroove disposed so as to be opposed to the rolling-element rollinggroove of the guide rail, the slider moving relatively in the axialdirection of the guide rail with rolling elements rollably interposedbetween the two rolling-element rolling grooves, and a crowning portiondisposed on each of the axial ends of the rolling-element rolling grooveof the slider, wherein a maximum value of a total load in an axialdirection applied on the crowning portions on the axial ends of theslider by the rolling elements is set to be smaller than a maximum valueof load in the axial direction applied on one of the crowning portionsof the slider by one rolling element.
 2. A linear guide apparatuscomprising: a guide rail having a rolling-element rolling groove formedon a side thereof along in an axial direction; a slider having arolling-element rolling groove disposed so as to be opposed to therolling-element rolling groove of the guide rail, the slider movingrelatively in the axial direction of the guide rail with rollingelements rollably interposed between the two rolling-element rollinggrooves, and a crowning portion disposed on each of the axial ends ofthe slider, wherein the crowning portion is a linear crowning having aconstant inclination angle θ and satisfying the following relationship:0<θ(N×t−Le)<δ₀ wherein N represents an integer of 1 or more; trepresents the distance between centers of the rolling elements (withseparator) or a diameter of the rolling element (free of separator); Lerepresents the length of the non-crowning portion; and δ₀ representspreload amount (diameter of rolling element based on the diameter ofrolling element which gives zero elastic deformation at the non-crowningportion).
 3. A linear guide apparatus comprising: a guide rail having arolling-element rolling groove formed on a side thereof along in anaxial direction; and a slider having a rolling-element rolling groovedisposed so as to be opposed to the rolling-element rolling groove ofthe guide rail, the slider moving relatively in the axial direction ofthe guide rail with rolling elements rollably interposed between the tworolling-element rolling grooves, and a crowning portion disposed on theboth axial ends of the slider, wherein the crowning portion is an arccrowning having a constant radius R and satisfying any of the followingrelationships (1) to (3): Nt−Le≦0.5t and 0.5{square root over (2Rδ₀)}<N×t−Le<1.5{square root over (2Rδ₀)};  (1) 0.5t<N×t−Le <1.5{squareroot over (2Rδ ₀)}; and   (2) t<{square root over (2Rδ₀)}  (3) wherein Nrepresents an integer of 1 or more; t represents the distance betweencenters of the rolling elements (with separator) or the diameter of therolling element (free of separator); Le represents the length of thenon-crowning portion; and δ₀ represents preload amount (diameter ofrolling element based on the diameter of rolling element which giveszero elastic deformation at the non-crowning portion).